Improving learning in mathematics
PD2: Learning from mistakes and
misconceptions
Aims of the session
This session is intended to help us to:

reflect on the nature and causes of
learners’ mistakes and misconceptions;

consider ways in which we might use
these mistakes and misconceptions
constructively to promote learning.
Assessing learners’ responses



Look at the (genuine) examples of learners'
work.
Use the grid sheet to write a few lines
summarising:
 the nature of the errors that have been
made by each learner;
 the thinking that may have led to these
errors.
Discuss your ideas with the whole group.
Saira: Fractions and decimals
Saira: Fractions and decimals
Saira: Fractions and decimals
Saira: Fractions and decimals

Confuses decimal and fraction notation.
1
(0.25 = 25 )

Believes that numbers with more decimal
places are smaller in value.
(0.625 < 0.5).
3

Sees 8 as involving the cutting of a cake
into 8 parts but ignores the value of the
numerator when comparing fractions.
Damien: Multiplication and division
Damien: Multiplication and division
Damien: Multiplication and division
Damien: Multiplication and division


Believes that one must always divide the
larger number by the smaller (4 ÷ 20 = 5).
Appears to think that:
 division 'makes numbers smaller’;
 division of a number by a small quantity
reduces that number by a small
quantity.
Julia: Perimeter and area
Julia: Perimeter and area
Julia: Perimeter and area
Julia: Perimeter and area



Has difficulty explaining the concept of
volume, which she describes as the
'whole shape.'
Believes that perimeter is conserved
when a shape is cut up and reassembled.
Believes that there is a relationship
between the area and perimeter of a
shape.
Jasbinder: Algebraic notation
Jasbinder: Algebraic notation
Jasbinder: Algebraic notation
Jasbinder: Algebraic notation




Does not recognise that letters represent
variables. Particular values are always
substituted.
Shows reluctance to leave operations in
answers.
Does not recognise precedence of
operations: multiplication precedes
addition; squaring precedes multiplication.
Interprets '=' as 'makes’ ie a signal to
evaluate what has gone before.
Why do learners make mistakes?

Lapses in concentration.

Hasty reasoning.

Memory overload.

Not noticing important features of a problem.
or…through misconceptions based on:

alternative ways of reasoning;

local generalisations from early experience.
Generalisations made by learners

0.567 > 0.85
The more digits, the larger the value.

3÷6 = 2
Always divide the larger number by the smaller.

0.4 > 0.62
The fewer the number of digits after the decimal
point, the larger the value. It's like fractions.

5.62 x 0.65 > 5.62
Multiplication always makes numbers bigger.
Generalisations made by learners

1 litre costs £2.60;
4.2 litres cost £2.60 x 4.2;
0.22 litres cost £2.60 ÷ 0.22.
If you change the numbers, you change the
operation.

Area of rectangle
≠ area of triangle
If you dissect a shape and
rearrange the pieces, you
change the area.
C
B
C
A
B
A
Generalisations made by learners

If x + 4 < 10, then x = 5.
Letters represent particular numbers.

3 + 4 = 7 + 2 = 9 + 5 = 14.
‘Equals' means 'makes'.

In three rolls of a die, it is harder to get 6, 6, 6
than 2, 4, 6.
Special outcomes are less likely than more
representative outcomes.
Some more limited generalisations


What other generalisations are only true in
limited contexts?
In what contexts do the following
generalisations work?






If I subtract something from 12, the answer will be
smaller than 12.
The square root of a number is smaller than the
number.
All numbers can be written as proper or improper
fractions.
The order in which you multiply does not matter.
You can differentiate any function.
You can integrate any function.
What do we do with mistakes and
misconceptions?

Avoid them whenever possible?
"If I warn learners about the misconceptions as I
teach, they are less likely to happen.
Prevention is better than cure.”

Use them as learning opportunities?
"I actively encourage learners to make mistakes
and to learn from them.”
Some principles to consider

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Encourage learners to explore misconceptions
through discussion.
Focus discussion on known difficulties and
challenging questions.
Encourage a variety of viewpoints and
interpretations to emerge.
Ask questions that create a tension or ‘cognitive
conflict' that needs to be resolved.
Provide meaningful feedback.
Provide opportunities for developing new ideas
and concepts, and for consolidation.
Look at a session from the pack



What major mathematical concepts are involved
in the activity?
What common mistakes and misconceptions will
be revealed by the activity?
How does the activity:
 encourage a variety of viewpoints and
interpretations to emerge?
 create tensions or 'conflicts' that need to be
resolved?
 provide meaningful feedback?
 provide opportunities for developing new
ideas?